A Capstone Course for Future Teachers of Secondary Mathematics

1. A Capstone Course for Future Teachers of Secondary Mathematics Development and Implementation Lisa Rezac and Melissa Shepard Loe University of St. Thomas Ninth Annual AMTE Conference January 2005

2. Presentation Outline • Motivation and Justification (MET, PMET) • Implementation: • Course catalog description • Syllabus and Course Topics • Bridges Through Number Theory • Assessment Topics • Exposition Project: Historical research • Exploration Project: Contemporary research • Discrete Mathematics • Topics in “Higher Mathematics” • Triumphs, Disappointments, Difficulties

3. Motivation and Justification: • CBMS: Conference Board of the Mathematical Sciences • MET: Publication from CBMS: The Mathematical Education of Teachers (Parts I and II) published by the MAA • PREP/PMET: Professional Enhancement Programs of the MAA and Preparing Mathematicians to Educate Teachers • MNBoT: MN Board of Teaching

4. MET Document Observation “There is evidence of a vicious cycle in which too many prospective teachers enter college with insufficient understanding of school mathematics, have little college instruction focused on the mathematics they will teach, and then enter their classrooms inadequately prepared to teach mathematics to the following generations of students.”

5. MET Document Recommendation “… mathematics departments can support the design, development, and offering of a capstone course in which conceptual difficulties, fundamental ideas and techniques of high school mathematics are examined from an advanced standpoint.”…”Such a capstone sequence would be most effectively taught through a collaboration of faculty with primary expertise in mathematics and faculty with primary expertise in mathematics education and experience in high school teaching.”

6. Our Philosophy: Capstone courses should… • Be taken after most math content courses are completed, so students have a solid background of traditional mathematical knowledge. • Build bridges between the content in the required college courses and fill in gaps from within the courses. • Maintain a high level of mathematical content and rigor and also address related pedagogical issues. • Most importantly, connect college level math courses and the high school curriculum.

7. ImplementationAdvanced Mathematics: Exploration and Exposition Description from the Course Catalog: This course gives students a sense of the history, applicability and currency of one or more mathematical ideas and serves as a capstone mathematics course for students seeking to teach secondary mathematics. In the course, students make substantial oral and written presentations on topics carefully selected to have a strong relationship to secondary school mathematics. They use publications, e.g. The American Mathematical Monthly, Mathematics Magazine, Mathematical Intelligencer and Scientific American, as well as standard texts as sources for their work.

8. Partial Syllabus Title: Math 450 Advanced Mathematics: Exploration and Exposition Credits: 4 semester credits (one regular course) Required: Of Math majors with a Secondary Ed. Co-major. Others welcome, will count towards a general math major. Goals: Provide a bridge between upper level math courses Explore connections to the high school curriculum More – related to NCTM standards and MN state standards Assessment: Two individual projects, homework, final exam, class participation, occasional quizzes, class discussions Topics/Units: Bridges Through Number Theory Assessment Topics Exposition: Historical research Exploration: Contemporary research Discrete Mathematics Topics in “Higher Mathematics”

9. I. Bridges through Number Theory • Using Simon Singh’s book, Fermat’s Enigma, we study the history of a mathematical theorem, practice reading mathematics, and look at the connections between abstract algebra, number theory and analysis. • Other possible choices could include • The Crest of the Peacock by George Gheverghese Joseph • Four Colors Suffice by Robin Wilson • The Code Book by Simon Singh • The Music of the Primes by Marcus Du Sautoy • For more possibilities, see handout which lists sources and reviews.

10. II. Assessment Topics • Assessing student work: • Solomon Friedberg, The Boston College Mathematics Case Studies Project • Bonnie Gold et al, Assessment Practices in Undergraduate Mathematics. • Katherine Merseth, Windows on Teaching Math • Assessing Texts: reviewed the idea of mathematical definitions by specifically looking at the idea of “limit” in many different Calculus texts. Traced the MVT, Rolle’s Theorem, and the Fundamental Theorem of Calculus through the same set of texts.

11. III. Exposition Project: Historical Research Individual presentation and paper on a historical search of a mathematical idea, person or theorem through history. Must include a discussion of how the topic could be used in a high school classroom with reference to MN State Standards for curriculum. Their model is the “Bridges” study of Fermat’s Last Enigma.

12. Exposition Project Titles • A Wonder Tool Lost Forever (slide rule) • Leonhard Euler: His Life and Contributions to Mathematics • African American Women in Math • Georg Cantor and the infinite • Fractal Geometry and the Mandelbrot Set • Geometric Series, Descartes, and the fundamental theorem of calculus • Hyperbolic Geometry • Sophie Germain and her Mathematics • Game Theory and the Prisoner’s Dilemma • A Slice of Pi, Pi and e • Map-Coloring and the Four Color Theorem • Fundamental Theorems of Algebra and Calculus • Symbolic Logic • Trigonometry • Pigeonhole Principle and Ramsey Theory

13. Student Project Rubrics • The first time the course was run, students helped create the rubric(s). • Rubric was handed out with the project assignment so they knew how the instructor would be assessing them. • These rubrics were also used by all the students in the class to assess the presentations. Comments were typed and used as anonymous feedback for the presenters. • See actual rubrics in the handout.

14. IV. Exploration Project: Contemporary Research Individual exploration of a current research article in The College Mathematics Journal or Mathematics Magazine and presentation in a seminar/colloquium style. Also must include a discussion of how the topic might be used in a high school classroom.

15. Exploration Articles(more on handout) • Pillow Chess, by Grant Cairns, Mathematics Magazine vol. 75, no. 3, June 2002, pages 173-186. • A visit With Six, by Monte Zerger, College Mathematics Journal, vol. 33, no. 2, March 2002, pages 74 – 87. • A Natural Generalization of the Win-Loss Rating System, by Charles Redmond, Mathematics Magazine, vol. 76, no. 2, April 2003, pages 119 – 126.

16. V. Discrete Mathematics • Accessible to H.S. and college students, NCTM Standard: Number and Operation • Other NCTM Standards: Problem Solving, Reasoning and Proof, Communication • Univ. of St. Thomas and MN Board of Teaching standards require discrete mathematics • Rich source of problems for H.S., college, and “recreational mathematics”

17. VI. Topics in “Higher Mathematics” Connected to Problems in Secondary Mathematics • Topics: induction, recursion, divisibility • Text: Usiskin et al, Mathematics for High School Teachers: An Advanced Perspective

18. Triumphs: • Student project work and presentations showed definite improvement from first to second projects. • Some students who were average students in the typical classroom showed their true teaching potential in their projects. • Communications with School of Education increased. • Awareness / appreciation of the work involved in doing mathematics. Students experienced working on mathematics with classmates, and gave brief presentations to explain solutions. • 7 of the 8 students in Fall ‘04 volunteered at a regional NCTM conference in Mpls 2004, got exposed to the “movers and shakers” in NCTM – better chance of continued participation at such conferences.

19. Disappointments: • Students unprepared for and unhappy with second project requirements. Although they were warned that this would be the hardest part of the course, and that they needed to begin their research during the first week of classes, this was apparently not taken seriously. • Connections to the High School curriculum need to be more explicit. • Heavy homework grading – big time demand. • Procrastination - Some students tended to delay starting work on the major projects until a week or so before due date. Result – low quality work, stress, less attention to other course work for 450!

20. Difficulties • Final Exam depends on student projects, needs careful coordination. • How can I call it a capstone course when I must cover new content? • How can I have a better connection with the content of the high school curriculum? • Motivating students to eliminate procrastination and set (and meet!) intermediate goals for projects. • Helping students in managing multiple assignments / projects concurrently. Helping students to see the course as a tapestry of experiences and learning as opposed to a collection of independent disconnected components.

21. Student Descriptions of their own Persistent Mathematical Difficulties Possible opening activity for the course: In looking over your entire mathematical career, identify three mathematical concepts that seemed important, yet you had a hard time understanding them. Describe the specific difficulties that you had or have, and if you have overcome them, how and when. You may also discuss a concept that you have not yet understood or have given up trying to understand.

22. Self-reported Student Difficulties

23. Discussion of Student Difficulties • Students still made mistakes when describing concepts they thought they had overcome. • Students who could trace their difficulties through several different applications did a good job of explaining concepts in an abstract way.

24. Conclusion We have discussed the motivation, creation, implementation and results of a capstone course for future teachers of secondary education. We hope to continually work on improving this course and would like to spend some time as a group on any questions or comments you may have. We provide again our beliefs on capstone courses as a set of possible discussion points:

25. Our Philosophy: Capstone courses should… • Be taken after most math content courses are completed, so students have a solid background of traditional mathematical knowledge. • Build bridges between the content in the required college courses and fill in gaps from within the courses. • Maintain a high level of mathematical content and rigor and also address related pedagogical issues. • Most importantly, connect college level math courses and the high school curriculum.

26. CONTACT INFORMATION Melissa Shepard Loe: mashepard@stthomas.edu Lisa Rezac: lmrezac@stthomas.edu University of St. Thomas St. Paul, MN